Group Action notation and understanding

Question

I've been studying Group Theory for some time, but I still do not understand the appeal of the group action notation. I mean - every time I see: $a \cdot e$, I am thinking, why not just treat $a$ as function on X, and just write $e(a)$ and also instead of $a \cdot gh$, just write $h\circ g(a)$? I mean we already have a symbol for function composition. (We have to reverse the order in this case, right?)

My understanding is, that when $G$ acts on $X$, we are saying that we have a homomorphism from $G$ to a subset of $\operatorname{Sym}(X)$. So why not just treat the elements of $G$ as we would treat their images in $\operatorname{Sym}(X)$? That is, just think of $f: G \to \operatorname{Sym}(X) $ and then treat the elements of $G$ as we would treat $\ker(f)g$.

I guess there are some reasons, why things are done as they are, so hopefully someone more knowledgeable can help.

Answer 1

In a sense, "$g(x)$" is the ancestor of "$g\cdot x$", when $g$ could be only a bijection on $X$ (then more commonly denoted with an $f$ or a $\sigma$). The notion of abstract group action is patterned upon the natural action of the set (forget for now about "groups") of all the bijections on a given set $X$, on this same latter set. Such a natural action, namely $(\sigma,x)\mapsto \sigma(x)$, fulfils the following two basic properties:

1. $Id_X(x)=x, \space\forall x\in X$;
2. $(\sigma\tau)(x)=\sigma(\tau(x)), \space\forall \sigma,\tau\in\operatorname{Sym}(X),\forall x\in X$.

Soon after that $\operatorname{Sym}(X)$ has provided the template to draw the definition of abstract group $G$ (the four axioms: closure, associativity, identity, inverse elements), one can be driven by 1 and 2 above, and rise up the following question: does anything interesting come from considering any map "$\cdot$" $: G\times X\to X$ such that (mimicking 1 and 2 above):

• $e\cdot x=x, \space\forall x\in X$;
• $(gh)\cdot x=g\cdot(h\cdot x), \space\forall g,h\in G, \forall x\in X$

? In such a more abstract scenario, "$g(x)$" doesn't seem quite sensible, since $g\in G$ isn't any longer, in general, a bijection on $X$.

Answer 2

This notation is also used for deterministic automata, which are just an action of the free monoid on a set $X$. Graphically, composition becomes visually natural with this convention. Just compare the following two lines: \begin{align} x \xrightarrow{g} x\cdot g \xrightarrow{h} (x\cdot g) \cdot h \\ x \xrightarrow{gh} x\cdot gh \qquad \end{align}